A Superelliptic Equation Involving Alternating Sums of Powers
نویسنده
چکیده
In this short note, we solve completely the Diophantine equation 1 − 3 + 5 − · · · + (4x− 3) − (4x− 1) = −y, for 3 ≤ k ≤ 6. This may be viewed as a “character-twisted” analogue of a classic equation of Schaffer (in which context, it was previously considered by Dilcher). In our proof, we appeal primarily to techniques based upon the modularity of Galois representations and, in particular, to a combination of these ideas with suitable local information.
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For n, k ∈ Z≥0, let Tn(k) be the alternating sums of the n-th powers of positive integers up to k − 1: Tn(k) = ∑ k−1 l=0 (−1)l. Following an idea due to Euler, we give the below formula for Tn(k): Tn(k) = (−1) 2 n−1 ∑ l=0 (n l ) Elk n−l + En 2 (
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تاریخ انتشار 2011